ON THE MORAVA K-THEORY OF SOME FINITE 2-GROUPS. Björn Schuster

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1 ON THE MORAVA K-THEORY OF SOME FINITE 2-GROUPS Björn Schuster Abstract. We compute the Morava K-theories of finite nonabelian 2-groups having a cyclic maximal subgroup, i.e., dihedral, quaternion, semidihedral and quasidihedral groups of order a power of two. 1. Introduction For any fixed prime p and any nonnegative integer n there is a 2(p n 1)-periodic generalized cohomology theory K(n), the n th Morava K-theory. Let G be a finite group and BG its classifying space. For some time now it has been conjectured that K(n) (BG) is concentrated in even dimensions. Standard transfer arguments show that a finite group enjoys this property whenever its p-sylow subgroup does. It is easy to see that it holds for abelian groups, and it has been proved for some nonabelian groups as well, namely groups of order p 3 ([7]) and certain wreath products ([3], [2]). In this note we consider finite (nonabelian) 2-groups with cyclic maximal subgroup, i.e., dihedral, semidihedral, quasidihedral and generalized quaternion groups of order a power of two. Theorem. Let G be a nonabelian 2-group with cyclic maximal subgroup and n > 1. Then the Morava K-theory of BG is muliplicatively generated by three classes in dimensions 2, 2 and 4, respectively, modulo some explicit relations. The Morava K-theories of the dihedral and quaternion group of order eight have been known for a while (see [7], with a correction in [8]); the relations mentioned in the statement of the theorem are written down explicitly in section Preliminaries The theorem is proved by calculating a spectral sequence of Atiyah-Hirzebruch- Serre type ( AHSSS ). For any extension of finite groups H G K and any generalized cohomology theory h, there is a spectral sequence (see [5]) E 2 = H (BH; h (BK)) = h (BG), 1991 Mathematics Subject Classification. 55N20, 55T99. Key words and phrases. Classifying spaces, Morava K-theory. Typeset by AMS-TEX 1

2 2 BJÖRN SCHUSTER natural both in the cohomology theory and in the extension. It generalizes the Atiyah-Hirzebruch as well as the Serre spectral sequence: for H = 1 it degenerates into the Atiyah-Hirzebruch spectral sequence ( AHSS ) for BK, whereas for h being ordinary cohomology one has the Serre spectral sequence. It is well known (see e.g. [9]) that the Atiyah-Hirzebruch spectral sequence for K(n) has first differential ( AHSS-differential ) d 2(p n 1) = v n Q n, where Q n is Milnor s operation defined inductively by Q 0 = β (resp. Sq 1 ) and Q i = [Q i 1, P pi ] (resp. Q i = [Q i 1, Sq 2i ] if p = 2); this fact will be used repeatedly below. 3. Proof of the theorem The proof splits into two cases. The first case covers dihedral, semidihedral and generalized quaternion groups: let G = < s, t s 2N+1 = 1, t 2 = s e, tst 1 = s r > where N 1, and either e = 2 N, r = 1 or e = 0, r = 1, 2 n 1. For e = 0, r = 1 the group G is the dihedral group D 2 N+2 of order 2 N+2, for e = 2 N, r = 1 we get the generalized quaternion group Q 2 N+2, while for N 2, e = 0 and r = 2 N 1 gives the semidihedral group SD 2 N+2. The center Z of G is a Z/2, generated by s 2N, with quotient isomorphic to the dihedral group D 2 N+1. Thus we get a central extension (3.1) 1 Z i G π D 2 N+1 1 (where D 4 = Z/2 Z/2). The second case are the quasidihedral groups QD 2 N+2, which can be presented as QD 2 N+2 = < s, t s 2N+1 = t 2 = 1, tst 1 = s 2N +1 > where N 2. The center of QD 2 N+2 is the cyclic subgroup generated by s 2 with quotient s, t isomorphic to Z/2 Z/2, i.e., we have a central extension (3.2) 1 Z/2 N i QD 2 N+2 π Z/2 Z/2 1. We shall compute the AHS spectral sequences of the extensions (3.1) and (3.2). We start with extension (3.1). The images s and t of s and t under π are generators for the quotient D 2 N+1. For N = 1, G is isomorphic to either the dihedral or the quaternion group of order 8; this computation has been carried out in [7]. However, since we will need some details of this computation in the course of this section, we want to summarize it briefly. Let a and b denote the multiplicative generators of H (B(Z/2 Z/2);F 2 ) dual to s and t. Then the extension class q is either a 2 + ab if G is dihedral or a 2 + ab + b 2 if G is quaternion. The E 2 -term of the spectral sequence can thus be identified with K(n) [z]/z 2n F 2 [a, b], and the only nontrivial

3 ON THE MORAVA K-THEORY OF SOME FINITE 2-GROUPS 3 differentials are d 3 and d 2 n+1 1, acting as d 3 z = a 2 b + ab 2, d 2 n+1 1a = v n a 2n+1 and d 2 n+1 1b = v n b 2n+1. The E 2 n+1-page is even dimensional, thus E 2 n+1 = E. There is also a nontrivial extension problem, see below. Now assume N > 1. Let K denote the subgroup of D 2 N+1 generated by s 2N 1 and t, T the subgroup generated by s 2N 1 and s t, and finally C = Z/2 N the cyclic subgroup generated by s. Then both K and T are isomorphic to Z/2 Z/2, and it is known that K and T detect D 2 N+1 in mod 2 cohomology (see [1], VI, 3). The inclusions of K, T, and C into D 2 N+1 produce three induced extensions (3.3) 1 Z G 8 K 1 with G 8 = D8 for G dihedral or semidihedral and G 8 = Q8 for G quaternion; (3.4) 1 Z G 8 T 1 with G 8 = D8 for G dihedral, G 8 = Q8 for G semidihedral or quaternion; and (3.5) 1 Z Z/2 N+1 Z/2 N 1 for any of the three types of G. We will use the associated maps of spectral sequences in the computations below. Recall the mod 2 cohomology of D 2 N+1 ([1], VI, 3): H (BD 2 N+1;F 2 ) = F 2 [x, y, w]/(x 2 + xy) where x, y H 1 (BD 2 N+1;F 2 ) = Hom(D 2 N+1,Z/2) are defined by x, s = 1, y, t = 1, x, t = y, s = 0, and w is the second Stiefel-Whitney class of the standard representation of D 2 N+1 in O(2). Moreover, the image of H (BD 2 N+1;F 2 ) in H (BK;F 2 ), H (BT;F 2 ) and H (BC;F 2 ) is given by the following table (see [4]): i K i T i C x y w 0 b a 2 + ab b b a 2 + ab u 0 v Here i K, i T and i C are the restrictions of H (BD 2 N+1;F 2 ) to the mod 2 cohomology of K, T and C, respectively, a and b denote the one dimensional polynomial generators of H (BK;F 2 ) (orh (BT;F 2 ) ) dual to s 2N 1 and t (or s 2N 1 and s t for T), u is the exterior generator of H (BC;F 2 ) in degree one and v its 2 N th Bockstein. The extension class q of (3.1) will play a significant role. It is given by (see [6]) w if G is dihedral w + y 2 if G is quaternion w + x 2 if G is semidihedral

4 4 BJÖRN SCHUSTER The spectral sequence calculation for (3.3) and (3.4) has been indicated above, and we have chosen our notation as to coincide with the one used there. We also want to sketch the calculation for (3.5). It is an almost immediate consequence of the fact that we know what the Morava K-theory of BZ/2 N+1 is, namely K(n) [ z]/ z 2(N+1) n with z in dimension two. Thus we have E 2 = Λ(u) F 2 [v] K(n) [z]/z 2n = K(n) (BZ/2 N+1 ) ; with the only nonzero differential being the one responsible for the truncation, i.e., d r(n) u = vn k(n) v 2Nn where r(n) = 2 Nn+1 1 and k(n) = (2 Nn 1)/(2 n 1). There is also a nontrivial extension, v n z 2n = v, and z represents the generator z of K(n) (BZ/2 N+1 ). We now return to the extension (3.1). The E 2 -term of its associated spectral sequence is E 2 = H (BD 2 N+1;F 2 ) K(n) (BZ) = F 2 [x, y, w]/(x 2 + xy) K(n) [z]/z 2n. Lemma 1. (a) d 3 z = yw. (b) z 2 is a permanent cycle. Proof. Let k(n) denote the connective analogue of K(n). We analyze first the differentials in the AHSSS converging to k(n) (BG) by comparing it to the mod 2- cohomology version: the Thom map k(n) HF 2 induces a natural transformation of cohomology theories and hence a map of spectral sequences. Let E 2 = H (BD 2 N+1;F 2 ) H (BZ;F 2 ) = F 2 [x, y, w]/(x 2 +xy) F 2 [u] H (BG;F 2 ) be the E 2 -term of the Serre spectral sequence in mod 2-cohomology, where u denotes the one dimensional generator of the fiber. In E r, u transgresses to q, hence u 2 = Sq 1 u to Sq 1 q. An easy argument shows that the map k(n) (BZ) = k(n) [z]/v n z 2n F 2 [u] = H (BZ;F 2 ) (we are abusing notation again by denoting z the generator for the connective Morava K-theory of BZ as well as for the nonconnective case) is given by z u 2 and v n 0. Therefore d 3 z Sq 1 q modv n in k(n)-theory, hence in K(n)-theory. (Note that d 2 is zero since the fiber is concentrated in even dimensions.) We want to show that this equation holds on the nose, not only mod v n. Suppose d 3 z = Sq 1 q +v n T for some class T E 3, v n 2. For dimensional reasons T has to be of the form z 2n 1 R, where R E 3,0 2 is a class in H 3 (bd 2 N+1;F 2 ). Then d 3 d 3 = 0 implies 0 = d 3 (Sq 1 q) + (d 3 z)v n z 2n 2 R + v n z 2n 1 d 3 R = d 3 (Sq 1 q) + v n z 2n 2 RSq 1 q + v 2 n z2n+1 3 R 2 + v n z 2n 1 d 3 R.

5 ON THE MORAVA K-THEORY OF SOME FINITE 2-GROUPS 5 As long as n > 1, z 2n+1 3 = 0 and comparison to the AHSS for BD 2 N+1 shows that d 3 is zero on classes not divisible by z, hence R = 0. (Recall that in the AHSS, d 2 n+1 1 is the first differential.) Now Sq 1 x 2 = Sq 1 y 2 = 0 and Sq 1 w = yw, which implies (a). Furthermore, let θ be the following two dimensional complex representation of G: θ(s) = ( ) e 0 0 e r, θ(t) = ( 0 ) 1 ( 1) σ 0 where e = exp(πi/2 N ), r = 1 if G is dihedral or quaternion, r = 2 N 1 if G is semidihedral, σ = 0 if G is dihedral or semidihedral and σ = 1 for G quaternion. Then θ(s 2N ) = I 2 (where I stands for the identity matrix), i.e., θ restricted to the center is twice the nontrivial representation η of Z/2. Thus the second Chern class of θ restricts to the class z 2 of the fiber. Note that yw is not a zero divisor in the cohomology of D 2 N+1 whence the E 4 - page remains a tensor product. Since x 2 = xy, all classes of the form x i w, i > 1, and y j w, j > 0 are eliminated; xw however is not. Also observe that the E 4 -term is detected by the E 4 -terms of the spectral sequences associated to (3.3) - (3.5). Thus the next potentially nonzero differential is d 2 n+1 1 = v n Q n, d 2 n+1 1x = v n x 2n+1, d 2 n+1 1y = v n y 2n+1. Detection by the spectral sequences of (3.3) - (3.5) also shows d 2 n+1 1w = 0 and d 2 n+1 1xw = (d 2 n+1 1x)w = x 2n+1 = 0. Finally, comparison to (3.5) shows that the remaining odd dimensional classes, xw k, are killed by the differential d r(n). More precisely, one has d r(n) xw = vn k(n) w 2Nn +1, where r(n) = 2 Nn+1 1 and k(n) = (2 Nn 1)/(2 n 1) as above. To obtain the ring structure of K(n) (BG) we still have to solve extension problems. This can be done, up to some indeterminacy, by comparison arguments once again. The indeterminacy arises from the fact that the E -page is no longer detected by the E -pages of (3.3) - (3.5), but the only classes eluding detection are those divisible by w 2Nn (for N = 1 the class ab plays the role of w in this respect). The filtration drop occuring for (3.5) was described above; for (3.3) and (3.4), that is the case N = 1, it can be worked out by restricting further to the three subgroups Z/2 of the base. Thus v n z 2n = { q + εvn (ab) 2n if N = 1 q + εv n w 2Nn if N > 1 where ε is either 0 or 1. Let y 1, y 2 and c 2 denote the classes represented by x 2, y 2 and z 2, respectively. (The reader should be warned that despite its suggestive name, c 2 is not the second Chern class of the representation θ, but only up to lower filtration. Thus the choice of c 2 is not canonical; this is the price we pay to obtain explicit

6 6 BJÖRN SCHUSTER relations.) Then K(n) (BG) = K(n) [y 1, y 2, c 2 ]/R where the relations R can be derived from the relations introduced in the spectral sequence by the differentials d 3 and d 2 n+1 1, and the filtration drop described above. Recall that q is either w, w + x 2 or w + y 2, depending on whether G is dihedral, semidihedral or quaternion, and note that it does not make a difference whether ε is 0 or 1, since at E, the class w 2Nn is annihilated by any class of positive horizontal degree. Thus we obtain the following list of relations (where we include the case N = 1 for completeness, see also [7] and [8]): (1) y 2n 1, y2n 2, v ny 1 c 2n 1 2 = v n y 2 c 2n 1 2 = y 1 y 2 = v 2 n c2n 2 if G = D 8 (2) y 2n 1, y 2n 2, v n y 1 c 2n 2 + y 2 1, v n y 2 c 2n 2 + y 2 2, v 2 nc 2n 2 + y y 1 y 2 + y 2 2 if G = Q 8 (3) y 2n 1, y2n 2, c2n 1 (2 Nn +1) 2, y 1 y 2 = y 2 1 (a) y 1 c 2n 1 2, y 2 c 2n 1 2 if G is dihedral (b) y v ny 1 c 2n 1 2, y v ny 2 c 2n 1 (c) y v ny 1 c 2n 1 2, y v ny 2 c 2n 1 if N > 1 and all three types, and 2 if G is quaternion 2 if G is semidihedral We now turn to extension (3.2), i.e., quasidihedral groups. We have E 2 = H (B(Z/2 Z/2); K(n) (BZ/2 N )) = F 2 [x, y] K(n) [z]/z 2Nn, with x and y dual to s and t, respectively, and z (as usual) the first Chern class of the standard complex character of Z/2 N. As before, we will use comparison arguments throughout the computation: let i 1, i 2 and i 3 denote the inclusions of s, st and t into s, t, respectively. Then we get three induced extensions 1 Z/2 N H k Z/2 1. H 1 and H 2 are cyclic of order 2 N+1, whereas H 3 = Z/2 N Z/2. Lemma 2. (a) d 3 z = (x 2 y + xy 2 )U where U is a unit in E 3 with d 3 U = 0. (b) z 2 is a permanent cycle. Proof. We start with (b). It suffices to show that there is a complex representation of QD 2 N+2 restricting to twice the standard representation on the fiber. Set θ(s) = ( ) e 0 0 e 2N +1, θ(t) = ( ) where e = exp(2πi/2 N+1 ). To prove (a), observe that z cannot be a permanent cycle: under the map induced by the natural transformation k(n) HF 2, z maps to the polynomial generator of H (BZ/2 N ;F 2 ), which has to transgress to the zero line (otherwise the dimension of H 3 (BQD 2 N+2;F 2 ) becomes too big). To actually compute d 3 z, we compare

7 ON THE MORAVA K-THEORY OF SOME FINITE 2-GROUPS 7 the spectral sequence of (3.2) to both the mod 2 cohomology spectral sequence as above and the Morava K-theory spectral sequences of the three extensions induced by i 1, i 2 and i 3. In every one of them, the only nontrivial differential is d 2 n+1 1; in particular, d 3 z restricts to zero on all three of them. Since the E 3 -term of (3.2) is a tensor product, d 3 z is a sum of products of classes in H 3 (BZ/2 Z/2;F 2 ) and elements of the Morava K-theory of the fiber of degree zero. Let αx 3 + βx 2 y + γxy 2 + δy 3 be an arbitrary class in H 3 (BZ/2 Z/2;F 2 ) and let u denote the one dimensional generator of H (BZ/2;F 2 ). The following table lists the images of x, y and αx 3 + βx 2 y + γxy 2 + δy 3 under the restrictions i k : i 1 i 2 i 3 x y αx 3 + βx 2 y + γxy 2 + δy 3 u 0 αu 3 u u (α + β + γ + δ)u 3 0 u δu 3 Now αx 3 + βx 2 y + γxy 2 + δy 3 restricts to zero under all three inclusions if and only if α = δ = 0 and β = γ = 1 or α = β = γ = δ = 0. In other words, d 3 z is divisible by x 2 y + xy 2. Together with the above observation that d 3 z is not zero in the cohomology spectral sequence, that is not zero modulo v n, this shows d 3 z = (x 2 y+xy 2 )(1+R), where R 1 is an element of degree zero in K(n) [z]/z 2Nn. Using d 3 d 3 = 0, we can conclude that there are no summands of R containing an odd power of z as a factor (since d 3 is nontrivial on odd powers one otherwise gets relations in the E 3 -term, contradicting its structure). Hence, 1+R is a cycle for d 3 and, since R is nilpotent, a unit in K(n) [z]/z 2Nn. As a consequence of part (a) of the lemma, zu 1 hits x 2 y + xy 2, so all the mixed powers of x and y of degree at least three become identified from the E 4 - page on. As in the previous calculations, E 4 is detected by the E 4 -pages of the spectral sequences of the extensions induced by i 1, i 2 and i 3 ; thus the next nontrivial differential is d 2 n+1 1. This differential kills all remaining odd dimensional classes, hence the AHSSS collapses at the E 2 n+1 1-page. Note that xy is a permanent cycle. The only remaining question is that of possible extension problems. Using comparison to the three induced extensions again yields vn k(n) z 2Nn = x 2 +εv n (xy) 2n where k(n) = (2 Nn 1)/(2 n 1) and ε = 0 or 1. Note that x 2 is the extension class of (3.2) in the sense that it is the class the exterior generator of the mod 2 cohomology of the fiber transgresses to in the mod 2 cohomology version of the spectral sequence. Finally, let y 1, y 2 and c 2 denote the classes represented by xy, y 2 and z 2, respectively; again, c 2 is a Chern class only modulo lower filtration. Then as a consequence of x 2 y = xy 2 and the filtration drop of z 2Nn we obtain the following relations: (1) y 2n +1 1, y2 2n, c2(n+1)n 1 2 (2) y1 2 = y 1y 2 = vn k(n) y 1 c 2Nn 1 2 = vn k(n) y 2 c 2Nn 1 2 where k(n) = (2 Nn 1)/(2 n 1).

8 8 BJÖRN SCHUSTER References [1] Z. Fiedorowicz and S. Priddy, Homology of Classical Groups over Finite Fields and Their Associated Infinite Loop Spaces, Lecture Notes in Mathematics 674, Springer-Verlag, Berlin, Heidelberg, New York. [2] M. J. Hopkins, N. J. Kuhn and D. C. Ravenel, Morava K-Theories of Classifying Spaces and Generalized characters for Finite Groups, Algebraic Topology, Barcelona 1990, Lecture Notes in Mathematics 1509, Springer-Verlag, Berlin, Heidelberg, New York, pp [3] J. R. Hunton, The Morava K-theories of wreath products, Math. Proc. Cambridge Phil. Soc. 107 (1990), [4] S. Mitchell and S. Priddy, Symmetric product spectra and splittings of classifying spaces, Amer. J. Math. 106 (1984), [5] D. C. Ravenel, Morava K-theories and finite groups, Contemp. Math. 12 (1982), [6] D. Rusin, The mod 2 cohomology of metacyclic 2-groups, J. Pure Appl. Algebra 44 (1987), [7] M. Tezuka and N. Yagita, Cohomology of finite groups and Brown-Peterson cohomology, Algebraic Topology, Arcata 1986, Lecture Notes in Mathematics 1370, Springer-Verlag, Berlin, Heidelberg, New York, pp [8] M. Tezuka and N. Yagita, Complex K-theory of BSL 3 (Z), K-Theory 6 (1992), [9] N. Yagita, On the Steenrod algebra of Morava K-theory, J. London Math. Soc. 22 (1980), Centre de Recerca Matemàtica, Apartat 50, E Bellaterra, Spain Current address: FB 7 Mathematik, Universität Wuppertal, Gaußstr. 20, Wuppertal, Germany address: schuster@math.uni-wuppertal.de